M is a positive integer ≥ 2 such that 3^M + 4^M is divisible by M.

Is M always divisible by 7?

If so, prove it. Otherwise, provide a counterexample.

(In reply to re: Possible Solution by Brian Smith)

My workings are long gone, but there is an additional note in my notebook that 3^M + 4^M  = (3+4)(other factors) if M is odd, as it must be.

My concern at that point seems to have been eliminating a 7-free M as a divisor of the (other factors) hence the digression into Fermat.

Posted by broll on 2017-06-24 23:37:45